Question

The demand and supply functions are p_d =1600 - x² and p_s = 2x² + 400 respectively. Find the consumer’s surplus and producer’s surplus at an equilibrium point.

Answer 1

At equilibrium p_d = p_s
1600 – x² = 2x² + 400
– x² – 2x² = 400 – 1600
– 3x² = – 1200
X² = `1200/3`
x² = 400
x = 20
If x = 20 p_o = 1600 – (20)²
= 1600 – 400
p_o = 1200
p_o x_o = 1200 × 20
= 24000

Consumer Surplus C.S.
= `int_(0)^(0)` px dx - `"p"_(0) "x"_(0)`

= `int_(0)^(0)` (1600- x²) - 24000

= `[1600x-"x"^3/3]_(0)^(20) - 24000`

= `[ 1600 (20) - (20)^3/3 - 24000]`

= 32000 - `8000/3` - 24000

= 8000 - `8000/3`

= `(24000-8000)/3`

= `16000/3`

C.S = 5333.3

Producer Surplus P.S.
`= x_0"p"_0 - int_(0)^(x_0)` g(x) dx

= 24000 -`int_(0)^(20)` 2x² + 400

= 24000 - `[(2"x"^3)/3 + 400x]`

= 24000 - `[(2(20)^3)/3+400(20)]`       

= 24000 - `[(2xx8000)/3+8000]`

= 24000 - `[16000/3+8000]`

= 24000 - 8000 - `16000/3`

= `(48000-16000)/3`

= `32000/3`

P.S. = 10666.67